Large N expansion of the 2-matrix model

TL;DR

This work develops a loop-equation framework to compute the full large $N$ topological expansion of the free energy for the two-matrix Hermitian model. By deriving a master loop equation and enforcing a genus-zero algebraic curve via rational uniformization, the authors obtain leading-order (genus zero) observables and then proceed to next-to-leading order, yielding an explicit genus-one free energy $F^{(1)}=-\frac{1}{24}\ln(\gamma^{4}D)$. The approach defines a rich set of loop functions and two-loop correlators, and provides algorithmic rules (pole-cancellation, interpolation) to determine higher-order terms, laying groundwork for systematic higher-genus computations and connections to 2D gravity and Ising-like matter on random surfaces. The results generalize known 1-matrix results, recover the ACM genus-one free energy in the appropriate limit, and offer a practical framework for exploring double-scaling limits and nontrivial matter coupled to gravity.

Abstract

We present a method, based on loop equations, to compute recursively all the terms in the large $N$ topological expansion of the free energy for the 2-hermitian matrix model. We illustrate the method by computing the first subleading term, i.e. the free energy of a statistical physics model on a discretized torus.